This improves upon an earlier result from [1]} which was only able to conclude that the power-law dimension was 0, and earlier results from Landrigan, which were only able to conclude that the \(\log \) -dimension was at most 1. There are two immediate consequences of this result: (i) the lower dimension bound shows that Landrigan's \(\log \) -dimension result is sharp, (ii) our upper dimension result reveals that there may be examples of operators with positive upper Lyapunov exponent that have a dimension strictly larger than \(1/\ln (1/t).\)
